Systems, such as industrial process systems, utilize advanced controllers to control processes carried out by the systems. Controllers are purchased by many companies having a wide variety of systems to control, from boilers and oil refineries to ice cream manufacturing, heating and cooling plants, and packaging plants to name a few. Such controllers utilize many parameters that are measured values corresponding to different pieces of equipment making up the system. The measured values may correspond to temperatures, pressures, flow rates, and many other measurable and estimated values for parameters that cannot be directly measured. Variables may be set by the controllers to control the processes. To help set the variables, the systems may be modeled using complex modelling techniques. The modelling may be referred to as system identification.
System identification is about science (algorithms) and art (experience). Both parts are indispensable when developing mathematical models. In a white-box model, an engineer knows the system/process, all parameter are known or measurable, and the model may be constructed using first-principles knowledge. In a gray-box model, some of the parameter are unknown and must be estimated using data. Physically well-founded models are available for use, but there is no systematic approach that may be used to estimate gray-box model parameters. In a black-box model, the engineer knows only experimental data and the model is built using some standard regression models. Some well-designed algorithms exist for parameter estimation of black-box models.
At present, more and more complex models are being developed as more complex systems are being controlled or observed. It is apparent that such models require more information to be correctly identified. The information can be collected by measuring experimental/operational data, but this can be intractable or expensive. On the other hand, a substantial piece of information comes from first-principle modeling. Using this information in model derivation results in more robust models with respect to errors in data, and it also lowers the demand on the amount of experimental data required.
A natural way to describe first-principle models is by using a gray-box modeling approach. The gray-box model (GBM) effectively captures user's knowledge about the system, which is typically being described by a set of parameterized differential equations. The GBM is parameterized by a vector of unknown parameters that are to be identified using experimental data.
Whereas the GBM definition falls into the “art” part of the system identification, the parameter estimation problem is about the “science.” Looking for the optimal value of GBM parameters results in a hard optimization task, which is the main drawback of the gray-box modeling approach.